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Combustion and Flame xxx (2014) xxx–xxx

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Combustion and Flame

journal homepage: www.elsevier .com/locate /combustflame

Tabulated chemistry approach for diluted combustion regimeswith internal recirculation and heat losses

http://dx.doi.org/10.1016/j.combustflame.2014.01.0150010-2180/� 2014 The Combustion Institute. Published by Elsevier Inc. All rights reserved.

⇑ Corresponding author. Fax: +33 (1)47028035.E-mail address: [email protected] (B. Fiorina).

Please cite this article in press as: J. Lamouroux et al., Combust. Flame (2014), http://dx.doi.org/10.1016/j.combustflame.2014.01.015

Jean Lamouroux a, Matthias Ihme b, Benoit Fiorina a,⇑, Olivier Gicquel a

a EM2C – CNRS, Ecole Centrale Paris, Châtenay Malabry, Franceb Department of Mechanical Engineering, Stanford University, Stanford, CA 94305, USA

a r t i c l e i n f o

Article history:Received 24 July 2013Received in revised form 9 January 2014Accepted 10 January 2014Available online xxxx

Keywords:Turbulent combustion modelingMILD (moderate or intense low oxygendilution)Flameless combustionTurbulent nonpremixed flamesLarge eddy simulationTabulated chemistry

a b s t r a c t

An efficient solution to reducing NOx formation is to maintain a relatively low flame temperature. Thiscan be achieved by mixing reactants, prior to combustion, with chemically inert diluents such as cooledcombustion products. In such diluted combustion systems, the flame temperature decreases because ofthermal ballast, limiting NOx production. This work focuses on modeling the specifics of this combustionregime in confined combustors. To characterize the dilution of reactants by burnt gases, the importanceof complex chemistry effects is emphasized and taken into account using a detailed chemistry tabulationapproach. This approach extends the flamelet/progress variable formulation by including informationabout the intensity of internal dilution rates and heat losses. A turbulent combustion model is then devel-oped in a large eddy simulation (LES) framework. The combustion model is validated by considering twocombustor configurations, namely an adiabatic burner and a combustor having isothermal walls – bothoperating under highly diluted combustion conditions. Simulation results are in good agreement withexperimental data, confirming the importance of detailed chemistry information and the validity ofthe tabulation approach to LES application to diluted combustion.

� 2014 The Combustion Institute. Published by Elsevier Inc. All rights reserved.

1. Introduction

The energetic efficiency of combustion systems can beenhanced by transferring heat from exhaust products to the freshgases by means of regenerative heating. Unfortunately, the result-ing increase in the reactant temperature has an adverse effect onthe formation of nitrogen oxides (NOx). An efficient solution toreducing NOx formation is to maintain a relatively low flametemperature. This can be achieved by mixing reactants with chem-ically inert diluents such as combustion products. In ‘‘diluted com-bustion’’ systems, the flame temperature decreases because ofthermal ballast, limiting NOx production. These systems promotedifferent combustion regimes such as MILD (moderate or intenselow-oxygen dilution) combustion [1,2], flameless oxidation [3]and high-temperature air combustion (HiTAC) [4,5].

In diluted combustion technologies, reactants are diluted withlarge amounts of burnt reaction products prior to combustion,which enables flame stabilization under lean conditions, therebyavoiding high-temperature regions that promote enhanced

thermal NOx formation. Experimental and numerical studies havebeen conducted in confined systems such as HiTAC combustionchambers [6,7], the IFRF (International Flame Research Founda-tion) semi-industrial-scale configurations [8–10], and reverse flowconfigurations [11,12]. These diluted combustion technologiesexhibits strong recirculation zones that enhance the mixingbetween fresh and burnt streams. Local stoichiometric conditionsare avoided, so that NOx production is dramatically decreased. Itis noteworthy that diluted combustion can be achieved withoutinternal recirculation of burnt gases: for example, Dally et al.[13] reproduced oxygen-diluted and MILD combustion regimes ina Jet in Hot Coflow (JHC) experiment.

A schematic of the model problem of diluted combustion withinternal product gas recirculation in a confined burner geometryis shown in Fig. 1. The schematic illustrates the modification inthe composition of reactants, which strongly varies with recircula-tion and dilution by burnt gases.

A major issue in the modeling of diluted combustion is thepronounced sensitivity of the flame structure to the reaction chem-istry [14,15]. To illustrate the importance of the detailed chemistrywhen a gas mixture is subjected to dilution by hot reactionproducts, a series of adiabatic laminar flames are computed. Thesesimulations consider chemical representations of increasing

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Nomenclature

LatinC normalized progress variableD molecular diffusivityH specific mixture enthalpySZ mixture fraction unmixedness factorT temperatureYc progress variableYD dilution variableYe elemental mass fractionYk species mass fraction of species kZ mixture fractionZ mixture fraction normalized with respect to undiluted

compositions

Greeka dilution parameterb heat loss parameterK progress parameterq density

/ equivalence ratio/G global equivalence ratiov scalar dissipationw vector of chemical quantities_x vector of chemical reaction rates, expressed in s�1

_xk chemical reaction rate of species k, expressed in s�1

SuperscriptsDil diluent streamF fuel streamOx oxidizer stream0 undiluted conditions

SubscriptsG global equivalence ratio conditioningja¼0 undiluted-conditioned quantityjb¼0 quantity evaluated without heat losses (b ¼ 0)jb¼1 quantity evaluated for maximal heat losses (b ¼ 1)

ass

frac

tion

[-]

0.015

0.02

0.025

0.03Detailed chemistry

1 step chemistry10% dilution

20% dilution

...

2 J. Lamouroux et al. / Combustion and Flame xxx (2014) xxx–xxx

complexity, namely the infinitely fast chemistry approximation, aone-step irreversible chemistry, and a detailed reaction sequence.The problem configuration consists of a steady-state counterflowmethane/air diffusion flame, and different values of strain rates(including the stable and the unstable branch) are considered. Bothreactant streams are diluted with burnt products, having a compo-sition that is identical to that of the MILD combustor configurationstudied by Verissimo et al. [16] (see Section 5.1). Chemical trajec-tories in CO2—CH4 composition space are presented in Fig. 2 for dif-ferent dilution levels. Results obtained with the detailed GRI 3.0mechanism [17] are shown by solid lines, the dashed line corre-sponds to the one-step irreversible chemistry, and the limiting caseof infinitely fast chemistry is shown by the symbol. From this fig-ure it can be seen that the detailed chemistry solution exhibits sev-eral possible trajectories, which depend on the level of dilution. Incontrast, the single-step chemistry trajectory is not sensitive to thedilution level, so the CH4 mass fraction exhibits a linear depen-dence on YCO2 . Since the infinitely fast chemistry model assumesthat the mixture is always at equilibrium state, the chemical trajec-tory reduces to a single point for a given value of equivalence ratio.Results from the detailed chemistry solution show that the chem-istry is affected by dilution, impacting fundamental flame proper-ties, including flame structure, species composition, and pollutantemission.

An attractive strategy for including detailed chemistry effectsusing moderate CPU resources are tabulated chemistry techniques

Fig. 1. Model problem of a diluted combustion configuration with internalrecirculation of burnt gases. Considering the idealized problem of methane/oxygencombustion, the initially separated reactants mix and react to produce CO2 andH2O. Reaction products are then recirculated where they modify the composition inthe oxidizer and fuel streams. The product gas dilution level is denoted by a.

Please cite this article in press as: J. Lamouroux et al., Combust. Flame (2014),

[18–22]. Among these, the flamelet model for nonpremixed com-bustion assumes that a turbulent flame can be decomposed intoa collection of one-dimensional flame elements [18]. Each flameletis then represented by a reaction–diffusion element that is con-structed between oxidizer and fuel streams. In the original formu-lation, proposed by Peters [18], the fuel and oxidizer compositionare assumed to be constant for each flame element. By construc-tion, this two-stream formulation is not able to account for effectsof reactant dilution by burnt gases on the chemical flame structure.To overcome this issue, a three-stream flamelet-progress variable(FPV) approach has recently been developed [23]. This modelwas applied to large eddy simulations of a Jet-in-Hot-Coflow(JHC) burner [23,24], in which the burner was operated in therecirculation-free adiabatic MILD operating regime. The dilution

CO2 mass fraction [-]

CH

4 m

0 0.02 0.04 0.06 0.080

0.005

0.01Equilibrium

Fig. 2. Chemical trajectories in CO2–CH4 state space for different dilution levels,illustrating the sensitivity of the fuel conversion to the reaction chemistry and thedilution. Here, the diluent is composed of the equilibrium product-gas compositionfor an equivalence ratio of / ¼ 0:58. Trajectories are extracted from steady laminarcounterflow diffusion flame computations, from the pure mixing line to the fullyburnt states.



Fig. 3. YCO evolution as a function of YCO2 during autoignition of a PSR for severalresidence times s. The solid trajectory corresponds to the 0-D reactor computation(with s ! 1 in Eq. (1)).

J. Lamouroux et al. / Combustion and Flame xxx (2014) xxx–xxx 3

is provided by a hot coflow stream, so the diluted reactant compo-sition does not vary in the combustion chamber. A model that usestwo independent mixture fractions is therefore sufficient to cap-ture the flame properties [23]. Nonetheless, when the dilution pro-cess is induced by strong internal recirculation, the localcomposition of the diluted fuel or oxidizer stream changes in timeand space, and the use of passive scalars to describe the mixturevariations is not adequate any more.

The present work presents a modeling route to account for theeffect of fresh-gas dilution by burnt products on the dynamics andstructure of turbulent flames. This approach is applied to the pre-diction of ‘‘diluted combustion’’ systems, including for instanceMILD (moderate or intense low-oxygen dilution) combustion[1,2] or flameless oxidation [3] regimes. To this end, a tabulationmethodology based on the FPV method, which accounts for effectsof internal dilution on the chemical flame structure, is first intro-duced. Since heat losses are important in most practical combus-tors, the impact of these effects on the chemistry will also beconsidered. Two tabulation strategies that differ in the representa-tion of heat transfer and dilution are considered. A sub-grid-scalemodel for the turbulence/flame interaction is proposed, and asemiempirical model is developed to represent the productionand internal recirculation of diluents. The capability of this LEScombustion model is demonstrated in two combustor configura-tions, namely an adiabatic burner and a combustor that utilizedisothermal wall treatment. These configurations were experimen-tally investigated at the Instituto Superior Técnico of Lisbon[25,16]. Both burners feature severe internal recirculation, verylow pollutant emissions, and high combustion efficiency. Compar-isons with experimental data are then carried out, and the validityof the proposed model is discussed.

2. Modeling of diluted combustion

2.1. Influence of fresh gas dilution on the chemical trajectories

The equilibrium thermochemical state is identified from theelement composition and the enthalpy h. Under unity Lewis num-ber assumption, the pair ðZ;HÞ, where Z is the mixture fraction andH the mixture enthalpy, is then sufficient to capture the equilib-rium state of the mixture, whether or not fresh gases have been di-luted by burnt gases. This is, however, not the case for reactionprocesses that are strongly affected by the rate of fresh gas dilu-tion. To illustrate this, we consider a 0-D constant-pressure perfec-tively stirred reactor (PSR), which is described by the following setof governing equations:

q@Yk

@t¼ q

sYb

k � Yk

� �þ q _xk

qCp@T@t¼ q

sXK

k¼1

YbkHb

k �XK

k¼1

YbkHk

!�XK

k¼1

q _xkHk ð1Þ

The inlet mass flow rate _m and the residence time s are related tothe volume V of the reactor through the relation _m ¼ qV=s. Thereactor is initially fed with a mixture of fresh gases at temperatureTðt ¼ 0Þ and composition Ykðt ¼ 0Þ. Under constant-pressure condi-tions, this mixture reaches the equilibrium state, denoted by super-script b. At t ¼ 0, gases of temperature Tb and composition Yb

k areinjected into the reactor and dilute the fresh gas mixture. As theelemental composition of the mixture is kept constant during thedilution process, the thermochemical equilibrium state is not af-fected. A series of PSR simulations are performed with the followinginitial gas composition and temperature: YO2 ¼ 0:14253;YCH4 ¼ 0:00214;YN2 ¼ 0:75633;YH2O ¼ 0:099, and T ¼ 1330 K, andusing the GRI 3.0 mechanism [17]. Figure 3 shows the solutions ofEq. (1) projected in the ðYCO;YCO2 Þ chemical subspace. Each curve


corresponds to a given residence time s of the gases inside the reac-tor. The solid line represents the chemical trajectory of a homoge-neous reactor (solution of Eq. (1) with s ! 1). For a longresidence time (s ¼ 100 ms), the chemical trajectory rapidly con-verges toward the closed reactor solution, and the fresh gases dilu-tion by burnt gases does not significantly affect the CO production.However, for shorter residence times (s ¼ 10 ms and s ¼ 1 ms), tra-jectories differs. The dimension of the attractive manifold increasesand a chemistry tabulation function limited to a single progress var-iable (here CO2) is not sufficient to capture the chemical processes.A solution to include dilution effects in the FPV model is presentedin the following section.

2.2. Chemistry tabulation

2.2.1. Flamelet/progress variable approachIn the FPV approach, the turbulent flame is locally seen as a col-

lection of one-dimensional laminar counterflow diffusion flames.Assuming steady combustion and neglecting preferential diffusioneffects, the structure of these flame elements is then described bythe solution of the flamelet equations, which are here written as

�vZ

2@2w

@Z2 ¼ _x; ð2Þ

where Z is the mixture fraction, w is a vector composed of all speciesmass fractions Y and temperatures T, and the vector _x denotes theirrespective source terms. vZ ¼ 2D j rZj2 is the scalar dissipation rateof the mixture fraction, which introduces the molecular diffusivityD of the mixture. Since the focus of this investigation is the dilutedcombustion of a methane/air mixtures, we assume equal diffusivi-ties for all species.

To accommodate variations in the reactant stream composi-tions, Eqs. (2) are supplemented by the following boundaryconditions:

wðZ ¼ 0Þ ¼ wOx; ð3ÞwðZ ¼ 1Þ ¼ wF; ð4Þ

where the superscripts F and Ox denote, respectively, boundaryconditions in the fuel stream and in the oxidizer stream. Thermo-chemical quantities are then functions of the mixture fraction andthe scalar dissipation rate. However, this parameterization doesnot ensure a unique thermochemical state space representation,since for given values of vZ , stable, unstable, and unburned states




may coexist. To overcome this issue, a reaction progress parameterK has been introduced [22,26], which provides a unique representa-tion of the entire thermochemical state space. The progress param-eter is defined through a progress variable Yc , which can beevaluated as the solution of a minimization problem [27]. A defini-tion for Yc in the context of methane/air combustion is

Yc ¼ YCO2 þ YCO þ YH2 þ YH2O: ð5Þ

Yc and Z are by definition statistically dependent. To facilitatethe estimation of the joint presumed probability density function(PDF) for the turbulent closure, the variable K is introduced. K isevaluated by conditioning Yc at a given value of the mixture frac-tion Z�,

K ¼ YcjZ� ; ð6Þ

and the thermochemical vector w is a function of mixture fractionand progress parameter:

w ¼ FwðZ;KÞ: ð7Þ

Since the balance equation for K is difficult to close for turbu-lent flows, it is preferable to solve a transport equation for the pro-gress variable Yc. The function F Yc , which relates Yc to Z and K, isidentified by using Eqs. (5) and (7):

Yc ¼ FYcðZ;KÞ: ð8Þ

Every thermochemical quantity contained in w evaluated in thesteady flamelet equations is therefore expressed as

w ¼ FwðZ;F�1YcðZ;YcÞÞ ¼ GwðZ; YcÞ: ð9Þ

1 Unlike [23], this dilution parameter is not a second mixture fraction. It tracks thedeviations, induced by dilution, of the chemical trajectories projected in a constant Zplane.

2.2.2. Diluted flamelet/progress variable approachCapturing dilution effects in a flamelet-based tabulated chemis-

try framework implies first modeling the mixing of the fresh reac-tants with the burnt product gases. As mentioned previously,assuming equilibrium state, the burnt gas composition and tem-perature depend only on the mixture fraction and enthalpy undera unity Lewis number assumption. These reaction products maymix at different rates with the fresh oxidizer and/or the fuelstreams, multiplying the number of possible configurations. Thedegrees of freedom underlying these processes are so numerousthat a direct tabulation of flamelets for all dilution configurationsis in practice not feasible. However, reasonable simplificationscan be made as in confined burners, the dilution is accomplishedby large-scale recirculation of burnt gases. Therefore, diluted com-bustion can be represented as a three-stream problem, consistingof a fuel stream, an oxidizer stream, and an internal reaction prod-uct gas stream. Because of large residence time in the recirculationzones, the burnt gas composition corresponds to the chemicalequilibrium of a fuel/oxidizer mixture that is characterized bythe global equivalence ratio /G. These large time scales also favorheat exchanges between recirculating burnt gases and burnerside-walls. Therefore, the diluent composition is in practice mainlya function of the global equivalence ratio and the level of wall heattransfer.

The study by Abtahizadeh et al. [28] showed that the dilutionprocess has a direct effect on the flame structure. In terms of chem-istry tabulation, accounting for the influence of diluent gradientsconsiderably increases the dimensionality of the tabulated mani-fold. In the approach developed here, we will neglect the influenceof these gradients on the flame structure.

By these assumptions, boundary conditions of the system of Eq.(2) are reformulated to account for the effect of dilution on the


flamelet archetype that is used for the chemistry tabulation. Thecounterflow flamelet boundary conditions now read

wðZ ¼ 0Þ ¼ wOxða; bÞ; ð10ÞwðZ ¼ 1Þ ¼ wFða; bÞ; ð11Þ

where a is the dilution parameter and b is the heat loss parameter.A schematic of representative flame-configurations and defini-

tions of the dilution and heat-loss parameters is shown in Fig. 4.The dilution parameter a is defined as the mass ratio between

fresh reactants and diluents: a ¼ 0 in the absence of diluents anda ¼ 1 when the mixture is saturated with diluents.1 To identify thecomposition of the diluent stream, we introduce YDil

k , denotingthe mass fraction of species k in the diluent stream. In practice,the diluent mixture composition YDil

k is evaluated from the thermo-chemical equilibrium state of the mixture under global equivalenceratio conditions. Denoting by YF

k and YOxk the mass fractions of species

k in the fuel and oxidizer streams, the definition of a reads

a ¼ YFk � YF;0

k

YDilk ðbÞ � YF;0

k

¼ YOxk � YOx;0

k

YDilk ðbÞ � YOx;0

k

; ð12Þ

and the superscript ‘‘,0’’ denotes the undiluted mixture.The heat loss parameter b is introduced to consider nonadia-

batic effects. This parameter is equal to zero under adiabatic con-ditions and b ¼ 1 when heat losses are maximum. By denoting asHDil the enthalpy (sum of chemical and sensible enthalpies) ofthe diluent, the definition of b is written as

b ¼HDil � HDiljb¼0

HDiljb¼1 � HDiljb¼0

: ð13Þ

From this definition, the composition and enthalpy of thediluted reactant streams (fuel and oxidizer) can be deduced:

YFkða; bÞ ¼ a YDil

k ðbÞ � YF;0k

� �þ YF;0

k ; ð14Þ

HFða;bÞ ¼ a HDilðbÞ � HF;0� �

þ HF;0; ð15Þ

YOxk ða;bÞ ¼ a YDil

k ðbÞ � YOx;0k

� �þ YOx;0

k ; ð16Þ

HOxða; bÞ ¼ a HDilðbÞ � HOx;0� �

þ HOx;0; ð17Þ

and HDil is written as

HDilðbÞ ¼ b HDiljb¼1 � HDiljb¼0

� �þ HDiljb¼0: ð18Þ

The minimal enthalpy of the diluent enthalpy HDiljb¼1 corre-sponds to the enthalpy of the diluent mixture that is cooled to aspecified minimum temperature TDil

min. In the following this temper-ature is associated with the burner wall temperature.

When all thermochemical quantities are parameterized interms of mixture fraction, progress variable, dilution parameter,and heat-loss parameter, w evolves in a four-dimensional subspace.In analogy to Eq. (9), this is written as

w ¼ GwðZ;Yc;a;bÞ; ð19Þ

and this formulation is referred to as diluted flamelet/progress var-iable (DFPV) approach.



Fig. 4. Schematic representation of the different constraints imposed on temperature and species of the boundary conditions used to generate counterflow diffusion flames;left: FPV model; right: DFPV model.


2.3. Relations between controlling scalars

All control parameters that appear in Eq. (19) are evaluatedfrom transported quantities. The mixture fraction Z is defined froma set of element mass fractions Ye [29]:

Z ¼ Ye � YOxe ðaÞ

YFeðaÞ � YOx

e ðaÞ: ð20Þ

Since the fuel and oxidizer elemental species mass fractions(YF

eðaÞ and YOxe ðaÞ respectively) are functions of the dilution ratio,

the balance equation for Z exhibits unclosed dependencies. Toavoid the dependency on the dilution parameter, we introduce amixture fraction formulation, which is defined with respect tothe undiluted mixture composition, and therefore independent ofa. This quantity is denoted by Z and has the following definition:

Z ¼ Ye � YOxe ja¼0

YFeja¼0 � YOx

e ja¼0

: ð21Þ

As such, Z can be directly evaluated by solving a conserved sca-lar transport equation. Furthermore, by introducing the mixturefraction ZG, which is evaluated under global equivalence ratio con-ditions (20) and (21) are related as

Z ¼ Z � aZG

1� a: ð22Þ

Figure 5 illustrates temperature profiles as a function of mixturefraction Z and different values of reaction progress Yc and dilutionlevels. In this configuration, the oxidizer consists of preheated airat 600 K and the fuel stream consists of methane at ambient tem-perature. The diluent is composed of burnt reaction products at anequivalence ratio of / ¼ 0:77 without any heat losses (b ¼ 0). Thechemistry is described by the GRI 3.0 mechanism [17]. The flamestructure is obtained from steady laminar flamelet computationsusing the FLAMEMASTER code [30]. This figure shows that the range

Fig. 5. Temperature profiles as function of mixture fraction Z for different levels of


in mixture-fraction space decreases with increasing dilution, untilit reaches a singular point that is defined by the pure diluentmixture.

The heat loss parameter, b, is linked to the enthalpy, H, which isexpressed in the Z-mixture fraction space as

HðZ;a;bÞ ¼ Z HFða;bÞ � HOxða; bÞh i

þ HOxða;bÞ: ð23Þ

Combining Eqs. (15), (17), and (22), and using the definition of bfrom Eq. (18), leads to

b ¼H � ðHF

a¼0 � HOxa¼0ÞðZ � aZGÞ � ð1� aÞHOx

a¼0 � aHDiljb¼0

aðHDiljb¼1 � HDiljb¼0Þ: ð24Þ

Note that this expression is valid for a – 0, which is consistentwith the underlying assumption that heat-loss and dilution effectsare coupled. Consequently, a ¼ 0 implies b ¼ 0.

For this approach, the rate of dilution needs to be estimatedfrom the evaluation of the reactant composition prior to combus-tion. Dilution with burnt gases as well as chemical reactions willaffect the progress variable Yc , defined from Eq. (5). To discrimi-nate the progress of reaction from dilution effects, the normalizedform of the progress variable, C, is introduced,

C ¼ Yc � Y0c ðZ;a; bÞ

Yeqc ðZ;a;bÞ � Y0

c ðZ;a;bÞ; ð25Þ

where Yeqc and Y0

c represent, respectively, the value of Yc at equilib-rium and in the unburned mixture diluted at a rate a.

The dilution variable Yd is defined from a combination of spe-cies mass fractions, chosen such that YF;0

d ¼ 0 and YOx;0d ¼ 0. The fol-

lowing relation between Yd and a is then obtained from Eq. (12):

a ¼ Yd

YDild ðbÞ

; ð26Þ

YDild ðbÞ denotes the value taken by the dilution variable Yd in pure

diluent, i.e. at equilibrium and for global equivalence ratio mixing

reaction progress and increasing levels of dilution levels a (from left to right).



Fig. 6. Steady flamelet solution at Z ¼ ZG for different dilution levels a showingtemperature as function of scalar dissipation rate. The operating conditionscorrespond to the MILD combustion experiment of [16]. Fuel is methane at ambienttemperature and air is preheated to 600 K. Global equivalence ratio is equal to/G ¼ 0:588 and no heat losses are considered.


conditions. In the following simulations the dilution variable is de-fined as Yd ¼ YCO þ YCO2 . From gas equilibrium computations, weobserved that this combination ensures a very weak dependencyof YDil

d on the heat-loss level b. The definition of a is then simplifiedto

a ¼ Yd

YDild ðb ¼ 0Þ

: ð27Þ

This simplification was tested for the experimental configura-tions and operating conditions investigated further, and was foundadequate for all mixture compositions. Yet this assumption shouldbe investigated further for application to heavy hydrocarbon fuels,which has not been considered in this study.

2.4. Analysis of laminar flame response

In the following section, a one-dimensional analysis is pre-sented to investigate the response of the flame structure to dilu-tion. This analysis provides a fundamental understanding aboutthe importance of considering dilution effects on the combustionprocess.

2.4.1. Dilution without heat-loss effectsFigure 6 shows simulation results for the S-shaped curve with-

out consideration of heat losses. As expected, dilution increases thetemperature of the unburned branch and the temperature of theburnt mixture converges to a unique equilibrium point. The criticalconditions for the quenching and ignition scalar dissipation ratesare also affected by dilution. With increasing dilution ratios, theextinction limit increases and eventually merges with the ignitionpoint. The transition between unburned and burnt conditions isthen smooth and monotonic, which is in accordance with the MILDcombustion regime characterization of Oberlack & Peters [2]. Last,combustion is sustained for high scalar dissipation rates if dilutionis sufficient: this explains why in most diluted combustion config-urations, combustion is stabilized despite high air injection veloc-ities. The influence of dilution on combustion is also highlighted bythe species trajectories in compositional space. Figure 7 shows theevolution of mass fractions of carbon dioxide and carbon monoxideas functions of Yc for different dilution rates and at a given mixturefraction.

From these results it can be seen that the chemical trajectoriesconverge to an attractor space that is defined by the undilutedmixture trajectory. As noted previously, the equilibrium composi-tion is identical for all cases, since the enthalpy of the mixture isnot affected by dilution. However, despite the convergence to the

(a)Fig. 7. Profiles of species mass fractions of (a) CO2 and (b) CO as fu


same manifold, these trajectories are strongly dependent on thedilution rate. The species production rates are also strongly depen-dent on the dilution. This is illustrated in Fig. 8b, where the pro-gress variable and the CO production rate are plotted asfunctions of Yc .

2.4.2. Dilution with heat-loss effectsIn practical burner configurations, heat losses play a crucial role

in the combustion process. Figure 9 presents various sets of S-curves obtained for different values of a and diluent temperaturesTDil. Independent of the level of heat losses applied to the diluent,the temperature of fresh gases increases with dilution. The equilib-rium temperature, obtained for low scalar dissipation rates, de-creases when TDil decreases and for increasing a. For each case,the unstable branch of the S-curve progressively vanishes withdilution.

The scalar dissipation rate value under quenching conditions,vq

Z , changes according to a and b. Without heat losses (case (a) inFig. 9), combustion is possible at very high strain rates if dilutionis sufficiently high: the increase in reactant temperature inducedby mixing with high-enthalpy diluent allows chemical reactionsto occur even under high-strain-rate conditions. However, for dil-uent temperatures below the adiabatic condition, two differentscenarios can be identified:

(b)nctions of Yc; conditions are identical to those used in Fig. 6.



(a) (b)Fig. 8. Profiles of species source terms of (a) Yc and (b) CO as function of Yc; conditions are identical to those used in Fig. 6.

(a)

(c)

(b)

(d)Fig. 9. Steady flamelet temperatures for Z ¼ ZG as function of the scalar dissipation rate for various dilution levels, a. Levels of increasing dilution are indicated by the arrow.Global equivalence ratio is equal to 0:77. (a) TDil ¼ TDil

b¼0 ¼ 2190 K; (b) TDil ¼ TDilb¼0:2 ¼ 1987 K; (c) TDil ¼ TDil

b¼0:4 ¼ 1766 K; (d) TDil ¼ TDilb¼0:6 ¼ 1530 K.


Please cite this article in press as: J. Lamouroux et al., Combust. Flame (2014), http://dx.doi.org/10.1016/j.combustflame.2014.01.015



� For low-heat-loss conditions, the trends are similar to the adia-batic dilution case; vq

Z increases with dilution, which favorscombustion by transferring thermal energy to the system.� If heat losses exceed a certain threshold, vq

Z decreases withincreasing dilution level. As result, the stable branch decreasesaccordingly.

In conclusion, the establishment of a MILD combustion regimeis dependent not only on the dilution with burned reaction prod-ucts but also on heat losses. While a pronounced dilution of reac-tants is necessary to achieve MILD combustion, heat losses are ofparamount importance in affecting the flame behavior and inextending the stability range in confined combustors with internalrecirculation.

3. Turbulent combustion model

3.1. Presumed probability density function model

A statistical description is used to represent the turbulence/chemistry interactions at the sub-grid-scale level. A presumedprobability density function (PDF) approach is considered, in whichthe filtered scalar quantities are written as

ew ¼ ZZZZ GwðZ;K;a; bÞePðZ;K;a;bÞdZdKdadb ð28Þ

and the density-weighted joint PDF eP is expressed as

ePðZ;K;a; bÞ ¼ qq

PðZ;K;a; bÞ; ð29Þ

with P being the nonweighted joint probability density function.The progress parameter K is defined to be statistically independentof the mixture fraction Z [31]. By assuming statistical independenceof a and b, the joint PDF can then be expressed in terms of marginaldistributions:

ePðZ;K;a; bÞ ¼ ePðZÞPðKÞPðaÞPðbÞ: ð30Þ

In the following, a beta-PDF is used to model the mixture frac-tion distribution, implemented following the approach of Lien et al.[32]. In MILD or highly diluted combustion regimes, the reactionzone is spatially distributed, which is the reason why it was oftencompared to well-stirred reactors [33,34]. As such, the distribu-tions of the reaction progress, dilution, and heat-loss variablesare represented by Dirac functions. After K is replaced by the pro-gress variable Yc , the following thermochemical state-space repre-sentation is obtained for the DFPV model:

ew ¼ eDwðeZ ; SZ ; eY c; �a; �bÞ: ð31Þ

Eq. (31) is used to provide information about all Favre-filtered ther-mo-viscous-chemical quantities (for example, temperature, species,source terms, density, and transport properties). It is noted that byintroducing Dirac distributions for the dilution and the heat lossparameters, we can simplify the notation and now write �a ¼ aand �b ¼ b.

3.2. Filtered transport equations

In addition to the solution of the conservation equations formass and momentum, five additional transport equations are re-quired to close the system of equations in the DFPV model. Here,scalar subgrid fluxes are modeled by introducing turbulent diffu-sivities DT;w, which are evaluated using a procedure for the dy-namic evaluation of the turbulent Schmidt numbers followingthe method of Lilly [35].


Transport equations for the first two moments of mixture frac-tion, progress variable, and enthalpy are written as

@�q eZ@tþr � ð�q~ueZÞ ¼ r � �qðDþ D

T;eZ ÞreZ� �; ð32Þ

@�qgZ002@t

þr � ð�q~ugZ002Þ ¼ r � �qðDþ DT;fZ002 ÞrgZ002

� �þ 2�qDT j reZ j2 � svZ

; ð33Þ@�qeY c

@tþr � ð�q~ueY cÞ ¼ r � �qðDþ D

T;eY cÞreY c

� �þ �q e_xYc ; ð34Þ

@�qeY d

@tþr � ð�q~ueY dÞ ¼ r � �qðDþ D

T;eY dÞreY d

� �þ �q e_xd; ð35Þ

@�qeH@tþr � ð�q~ueHÞ ¼ r � �qðDþ D

T;eH ÞreH� �: ð36Þ

The mixture-fraction subgrid scalar dissipation rate is modeledusing a linear relaxation assumption

svZ¼ 2�qCvDT

gZ002D2 ; ð37Þ

in which D is the LES filter size and Cv is a model constant taken asunity. Radiation effects in the enthalpy equation are neglected andonly convective heat losses to the combustor wall are considered.

To solve the balance equation for the dilution variable (35), thesource term e_xd needs to be modeled. This term accounts for inter-nal exchange of reaction products from the compositional streamto the dilution stream. Since this requires the consideration of mul-tispecies interaction, this term cannot be analytically related.Therefore, an empirical model for this transfer term is proposed.The role of the source term is to transform burnt gases into diluent.The source term peaks at the condition corresponding to the globalequivalence ratio. Whether or not the incoming streams will be-come diluted then depends on the convective velocity and the dif-fusivity. For this, we assume that burnt products areinstantaneously transformed into the diluent stream in the burntgas regions. Mathematically, this term is then written as

e_xd ¼1s|{z}I

eY Dild � eY d

� �|fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}

II

exp �eZ � ZG

r

!28<:

9=;|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}III

HðC � 1Þ|fflfflfflfflfflffl{zfflfflfflfflfflffl}IV

: ð38Þ

This expression consists of four terms that represent the follow-ing physical processes:

� Term I: This term is the inverse of the time needed to producediluents, which is here evaluated as s ¼ Dt, assuming instanta-neous transfer of products into the dilution stream.� Term II: This term relaxes eY d toward its equilibrium value. YDil

d

is dependent on Z and corresponds to the value of the productgas composition at equilibrium. Since Yd is linearly dependenton Z, it can be shown that YDil

d is given by:

http://

eY Dild ¼

~ZZG

Ydja¼1 for eZ 6 ZG;

Ydja¼1 �~Z�ZG1�ZG

Ydja¼1 for eZ > ZG:

8<: ð39Þ

� Term III: The assumptions made earlier state that the diluentmixture is composed of burnt gases that correspond to theproduct mixture at the global equivalence ratio. The Gaussianfunction included here, centered at ZG, localizes the productionof diluent in mixture fraction space and the value for its stan-dard deviation was chosen to be equal to r ¼ 0:05.� Term IV: HðC � 1Þ is the Heaviside function on C. The introduc-

tion of this function makes it possible to form diluents only inthe fully burned gas mixture.

dx.doi.org/10.1016/j.combustflame.2014.01.015



With the solution of Eqs. (32)–(34), the DFPV-state-variablescan then be evaluated as follows:

a ¼eY deY dja¼1

; ð40Þ

eZ ¼ eZ � aZG

1� a; ð41Þ

SZ ¼gZ002eZð1� eZÞ ¼

gZ002ðeZ � ZGÞð1� aþ aZG � eZÞ ; ð42Þ

b ¼eH � ðHF

a¼0 � HOxa¼0ÞðeZ � aZGÞ � ð1� aÞHOx

a¼0 � aHDiljb¼0

aðHDiljb¼1 � HDiljb¼0Þ: ð43Þ

4. Application to adiabatic MILD operating regime

To examine the performances of the DFPV model in applicationto an adiabatic MILD combustion regime, large eddy simulations ofa well-insulated combustion chamber are performed. The experi-mental configuration under consideration has been investigatedby Castela et al. [25]. The choice of using LES was motivated by pre-vious investigations by Graça et al. [36], in which it was concludedthat RANS computations, associated with EDC or transported PDFcombustion models, provide an inadequate description of the sca-lar mixing processes up to half the length of the combustionchamber.

In this investigation, we will compare the following three com-bustion models:

� The laminar flamelet model (LFM) of Peters [18], for whichonly the stable branch of the S-curve is tabulated, andextinction due to quenching is not captured.

� The FPV approach [22], which considers the solution of theunstable branch, but does not account for variations in thereactant composition.

� The DFPV model without heat losses (b ¼ 0) proposed here.

4.1. Experimental configuration

The configuration is a reversed-flow combustion chamber, inwhich inlet and exhaust ports are located on the same side. Thisconfiguration ensures sufficiently large residence times to com-plete combustion and to promote intense mixing of burned gaseswith the unburned reactant stream [37]. Similar reverse-flow con-figurations have previously been considered [11,12] in order toachieve MILD combustion conditions.

The configuration consists of a cylindrical combustion chamberof 340 mm length and 100 mm diameter. The injection system iscomposed of a central fuel nozzle with a diameter of 4 mm, supply-ing natural gas in the volumetric composition ratioCH4=C2H6=C3H8=N2 ¼ 85:1=7:6=1:9=5:4%. Preheated air at 600 Kis supplied through an annualar injector with an inner diameterof 14 mm and an outer diameter of 18.5 mm. The exhaust port isan annulus, having an inner diameter of 75 mm and an outer

Fig. 10. Schematic view of the combustion chamber geometry of Graça et al. [36],


diameter of 90 mm. A schematic of this configuration is shown inFig. 10.

The operating conditions investigated here are summarized inTable 1. The adiabatic operating condition was indirectly con-firmed by comparing computed adiabatic flame-temperature re-sults (at the global equivalence ratio) with reported exhausttemperature measurements, indicating that the measurementsare within 5% of the theoretical results. We thus considered thatthe configuration was adiabatic.

4.2. Computational setup

The finite-volume code YALES2 [38,39] was used to simulatethe combustor configuration. In this solver, the Navier–Stokesequations are solved under a low-Mach approximation using a pro-jection method [40] for variable density flows. For the presentedcomputations, the equations are solved in conservative form. Afourth-order finite-volume scheme is used to discretize the spatialoperators and a fourth-order Runge–Kutta-type scheme is used fortime advancement.

To investigate the sensitivity to the LES mesh resolution, simu-lations on three different computational grids were performed, andthe mesh characteristics are summarized in Table 2.

A detailed analysis of mean-flow results showed that speciesand temperature statistics were not further improved when themesh resolution was increased from the medium to the fine grid(Fig. 11). Therefore, only results from the medium mesh arepresented.

The lookup table was discretized with 201� 15� 201� 11 gridpoints in the direction eZ � SZ � C � �a. The generation of the filteredchemistry table was achieved in approximately 2 h on 12processors.

Inflow conditions for the oxidizer stream were prescribed froma turbulent velocity profile with 5% turbulent intensity. An analysisof sensitivity to the turbulent intensity was carried out, and negli-gible impact on the flow field was observed. As for the progressvariable, the boundary conditions values associated with the dilu-tion variable are set to zero for every inlet.

4.3. Instantaneous flow field

A comparison of instantaneous temperature field, obtainedfrom all three models is shown in Fig. 12. These qualitative resultsshow that the instantaneous temperature field, predicted by LFM,shows considerably higher temperatures than obtained from theother two models. The maximum temperature is close to the adia-batic temperature of 2150 K. This is due to the model formulation,which does not consider flame extinction (only the upper stablebranch of the S-curve is considered). This is further illustrated byshowing scatter plots of temperature on a longitudinal cut inFig. 13 (left). Since this model does not account for extinctionevents, all reactants mix and burn, and eventually relax towardthe equilibrium composition. Combustion is therefore consideredto be fast.

showing (left) a longitudinal cut and (right) a combustor cross-sectional view.



Table 1Global equivalence ratio /G, inlet velocity of air- and fuel streams, Uair and Ufuel ,respectively.

/G Uair (m/s) Ufuel (m/s) Reair Refuel sresG (s)

0:416 108 17;7 8751 4514 0:211

Note: Reynolds number based on air injector Reair and fuel injector Refuel, and globalresidence time sres

G evaluated as the ratio of the volume of the chamber to the totalreactant volume flow rate.

Table 2Mesh information for the simulation of the configuration of Castela et al. [25].

Mesh D0 (mm) DoF Elements

Coarse 0:4 1:2M 6MMedium 0:22 5:5M 28:5MFine 0:11 41M 228M

Note: D0 denotes the minimum filter size and DoF is degrees of freedom (inmillions).


The FPV approach allows for the presence of unburned mixturesas solutions of the flamelet equations (Fig. 12 (middle)). In thepresent case, the reactant gas temperature at the inlet is too lowto promote autoignition. Partial premixing of reactants occurs afterinjection into the combustion chamber. Reactants are then dilutedby recirculating hot burned gases, so that the flame stabilizationand combustion are facilitated by diffusion transfer of heat and

Fig. 11. Mean temperature and dry mole fractions of CO2 and O2 along t

Fig. 12. Temperature fields in a median plane for the three models. The distribution is shmodel; Middle: FPV model; Right: DFPV model.


radical species to the partially premixed reactant mixture. TheFPV model predicts a maximum temperature that is considerablylower than the LFM model. This is further illustrated by scatterplots in Fig. 13 (middle). Because of the partial premixing of thereactants, occurring prior to combustion, chemical trajectoriesconverge to the equilibrium condition at Z ¼ ZG, thereby bypass-ing the high-temperature regions around the stoichiometric condi-tion. It is noteworthy to point out that the FPV model predictscombustion of fuel-rich pockets, resulting in the formation of hotspots with temperatures around 1550 K; these spots are shownby the red areas in Fig. 12 (middle).

An instantaneous temperature field predicted by the DFPVmodel is illustrated in Fig. 12 (right), exhibiting qualitative similar-ities to the temperature field that is predicted by the FPV model(Fig. 12 (middle)). The detachment of the reaction zone from theregion between air and fuel inlets and temperature field is consid-erably lower than that obtained from the other two simulations.Figure 13 (right) shows that for the whole mixture fraction range,unburned gas states exist. Moreover, the penetration length of thereactants is lower in the DFPV model than in the FPV model, whichhighlights the effect of dilution on the combustion process.

4.4. Statistical flow field results

A comparison of time-averaged temperature profiles along thecombustor centerline is shown in Fig. 14. The temperature pre-dicted by the LFM model is overestimated until 170 mm. The FPV

he centerline of the configuration for the medium and fine meshes.

own close to the injection system and the spatial scale is given in meters. Left: LFM



Fig. 13. Scatterplots of temperature. The solution of an undiluted steady flamelet for vst ¼ 20 s�1 is shown by the dashed line. Left: LFM model; Middle: FPV model; Right:DFPV model.

Fig. 14. Time-averaged temperature along the centerline of the configuration.Symbols: experimental data points; Continuous line: DFPV model; Dashed line: FPVmodel; Dashed-dot-dot line: LFM model.


model predicts a lower heat release in the nozzle-near region, andgood agreement with experimental data is obtained forx > 150 mm, corresponding to 44% of the total length of thecombustion chamber. A noticeable improvement is obtained withthe DFPV model, where the temperature homogeneity is betterreproduced. It is still unclear why discrepancies arise in the noz-zle-near region, but it can be speculated that the mixing processrequires more precision, because of measurement uncertainties.Moreover, it is noteworthy that the discrepancies appearing inthe nozzle-near region are significantly lower than in the RANScomputations of Graça et al. [36], where the temperature at thecenterline at x ¼ 70 mm is overestimated by 800 K.

Similar trends are observed for dry mole fractions of O2 and CO2

(shown in Fig. 15). Considerable discrepancies can be observed for

Fig. 15. Comparisons of temporally averaged centerline profiles for mole fractionsof O2 and dry CO2. Symbols: experimental data points; Continuous line: DFPV model;Dashed line: FPV model; Dashed-dot-dot line: LFM model.


the predictions with the LFM model, while good agreement is ob-tained for the simulations using the FPV and the DFPV models. Thisprovides further support that the predicted flame characteristicsnear the injector are representative for this burner configuration.

Comparisons of carbon monoxide mole fractions along the cen-terline are shown in Fig. 16. Large differences are observed be-tween the three model predictions. The LFM model overpredictsthe reaction progress for all equivalence ratios, which leads to a ra-pid increase of CO in the nozzle-near region. This is followed by arapid consumption of CO due to the relaxation towards equilib-rium. The delayed combustion process predicted by the FPV modelyields an overprediction of CO. In contrast, when the dilution ofreactants is considered, the model qualitatively captures the trendand the magnitude of the CO profiles.

The accurate prediction of CO is particularly challenging, whichis due to its sensitivity to the reaction chemistry and localflow-field composition. The following comparison of CO resultsemphasizes the importance of considering dilution effects in thecombustion model. To illustrate the formation and consumptionof CO inside the combustor, we evaluate scatter data in CO–CO2

composition space throughout the combustor. These results arepresented in Fig. 17, confirming that dilution has a considerableinfluence on the CO conversion. When dilution effects areneglected, the only trajectory available is defined by the undilutedcase, and the production of CO is unsatisfactory. The resultsobtained from this large eddy simulation substantiate the laminarflame investigations and show that the flame structure and minor-species conversion exhibit pronounced sensitivity to the dilution ofthe reactant gases. Furthermore, these simulations demonstratethe capability of the proposed DFPV model in predicting internaldilution systems and dilution-controlled MILD combustion.

Fig. 16. Comparisons of temporally averaged centerline profile for dry molefractions of CO. Symbols: experimental data points; Continuous line: DFPV model;Dashed line: FPV model; Dashed-dot-dot line: LFM model.



Fig. 17. Scatterplots of CO as a function of CO2 (in dry mole fractions) foreZ ¼ ZG � 0:002. Black: DFPV; Red: FPV; Blue: LFM.

Fig. 18. Percentage of volume occupied by cells conditioned by the flame index GIZ .


4.5. Flame structures analysis

The Takeno index [41] is used to discriminate combustionmodes in the combustion chamber. Yet the formulation of thisindex poses issues when detailed chemistry effects are taken intoaccount and in the absence of reactants. To overcome thesedeficiencies, another flame index is proposed.

As mentioned in several studies [42–44], in moderately curveddiffusion flames, the gradients of Z (bnZ ¼ rZ= j rZ j) and Yc

(bnYc ¼ rYc= j rYc j) are aligned so that bnZ � bnYc � 1. For weaklystratified flames and premixed flames that are subjected to lowstrain, the flame front is perpendicular to iso surfaces of mixturefraction, which implies that bnZ � bnYc ¼ 0. In partially premixedcombustion, one can then obtain �1 < bnZ � bnYc < 1. In the follow-ing, we define the flame index GIZ as:

GIZ ¼reZ � reY c

j reZ jj reY c j

��e_xC>�

: ð44Þ

To ensure that we analyze only the reaction zone, this index isconditioned on the source term of the reaction progress variableand � is a parameter, which is here chosen to be equal to 1 s�1.GIZ ¼ �1 defines a diffusion-controlled combustion mode. It isimportant to note that Eq. (44) does not account for subgrid contri-butions. To assess the significance of subgrid contributions, weevaluated the flame index for the medium and the fine mesh,and both provide quantitatively similar results. The flame indexis computed from resolved quantities. It is expected that the qual-itative nature of the predicted flame structure is weakly sensitiveto the errors induced by the turbulent combustion model, even ifa feedback on the resolved quantities is expected. On tetrahedra-based meshes with high variations in the cell size, the distributionof GIZ needs to be weighted by cell volumes to estimate the volu-metric contribution of the different combustion modes that arepresent inside the burner. The discrete volume-weighted flame in-dex is written as

fZ ¼

PVcellje_xC>�;GIZP

Vcellje_xC>�

ð45Þ

The distribution of the volume-weighted flame index is illus-trated in Fig. 18. The distribution exhibits a bimodal shape, indicat-ing that combustion mainly occurs in the diffusion-controlledregime. A quantitative analysis showed that more than 55% ofthe reaction zone volume is characterized by eZ–eY C alignment an-gles in the range of �p=6;p=6½ � or �5p=6;5p=6½ � radians.


These results allow us to conclude that the combustion in thisreverse-flow combustor is primarily dominated by nonpremixedcombustion. However, a nonnegligible contribution to the reactionoccurs in gas pockets where GIZ differs from �1, indicating thepresence of premixed and partially premixed combustion regimes.This implies that the DFPV model is able to represent these flametopologies at strongly diluted MILD combustion regimes.

5. Application to a nonadiabatic MILD operating regime underconsideration of wall heat-loss effects

In the previous section, the applicability of the DFPV model forthe prediction of dilution-controlled combustion regimes withoutconsideration of heat-loss effects was examined. However, in prac-tical applications, heat losses play an important role in affectingthe combustion stability and species distribution. The perfor-mances of several models with different degree of complexity arehere evaluated in application to a nonadiabatic configuration:

� The FPV approach [22], which does not account for varia-tions in reactant composition and heat-losses. Note thatthe LFM model predictions are here omitted as they collapseon the FPV approach estimations.

� A simplified DFPV approach, denoted ‘‘sDPFV,’’ where dilu-tion and heat losses are correlated in time and space. Thisapproach implies a unique value of the dilution ratio a fora given enthalpy defect. This model is derived from theDFPV formulation, and modeling details are presented inAppendix A.

� The proposed DFPV model allows for compositional varia-tions as well as heat-loss effects without enforced correlations.

5.1. Experimental configuration

A schematic of the confined combustor configuration is pre-sented in Fig. 19. This burner was experimentally studied by Veris-simo et al. [16]. The cylindrical combustor has a diameter of 50 mmand a length of 50 mm. The reaction products exit through a con-vergent nozzle of length 150 mm and 15 degrees. Preheated airat 673 K is injected through a central nozzle, having a diameterof 10 mm. Methane fuel under ambient conditions is suppliedthrough 16 separated nozzle holes (2 mm in diameter), which arecoaxially located at a radius of 15 mm. The fuel inlet velocity isfixed at 6.2 m/s, and the experimental investigations consider only



Fig. 19. Schematic view of the combustion chamber geometry of Verissimo et al. [16] through a longitudinal cut (left), and a cross-sectional cut through the burner (right).

Table 3Global equivalence ratio /G, air inlet velocities Uair , Reynolds number based on airinjector Reair and global residence time sres

G .

Run /G Uair (m/s) Reair sresG (s)

2 0:77 113:2 17,526 0:294 0:58 143 22,140 0:23


variations in the air mass flow rate, thereby changing the overallequivalence ratio.

In the following, two operating conditions are considered,namely ‘‘run 2’’ and ‘‘run 4.’’ The corresponding operating condi-tions are summarized in Table 3. As reported in the experimentalstudy, high-momentum air injection increases the recirculationand the mixing of unburned reactants with reaction products. Thisresults in a homogeneous temperature field and ultralow emis-sions of CO and NOx. Verissimo et al. [16] studied the effect of ex-cess air on the combustion regime and reported a gradualdeviation from MILD combustion when the excess air coefficientwas increased. Probe measurements were conducted to determinespecies concentrations and the temperature field inside thecombustor. From the measurements, it was concluded that withincreasing excess air the flame structure shifts from ahomogeneous reaction zone toward a recirculating flame that sta-bilizes in proximity to the fuel injector.

The operating point denoted as ‘‘run 4’’ features the character-istics of a conventional lean combustion process. The authors ofthe experimental study reported a localized reaction zone and aclearly visible flame front. For the second case (designated ‘‘run2’’) the measurement indicates that the combustion occurs in theMILD combustion regime: the reaction zone is distributed over alarge volume and no flame front is visible. For both operating con-ditions, CO and NOx emissions are below 10 ppm. The measure-ment database includes mean flow results for temperature, O2,CO2, CO, NOx, and unburned hydrocarbons (UHC) along axial andradial sections in the burner. The reported reproducibility of theexperimental data is 5% for temperature and approximately 10%

for species measurements.

5.2. Computational setup

Simulations with two different grid resolutions have been per-formed (Table 4). A further grid refinement study was performedbut did not show noticeable influence on the scalar mean flowquantities. Therefore, only results obtained with the fine mesh of

Table 4Mesh information for the simulation of the configuration of Verissimo et al. [16].

Mesh D0 (mm) DoF Elements Nair

Coarse 0:4 2:58M 13:7M 25Fine 0:22 8:45M 48:9M 45

Note: D0 denotes the minimum filter size and DoF is degrees of freedom (in mil-lions); Nair is the number of grid points along a diameter of the air inlet.


48.9 million control volumes are presented in the following. Thefuel inlet velocity was prescribed by a laminar profile andturbulent inlet conditions were used for the air injection. Turbulentfluctuations with different turbulence levels (Tuinj ¼ 5% for run 4,and Tuinj ¼ 5% or 1% for run 2) have been superposed on the meanflow, which was evaluated from a periodic pipe flow simulation.From this study it was found that the mean velocity profile alongthe centerline is sensible to the turbulent intensity, but no exper-imental information about the velocity field was reported to guidethe simulation setup.

Figure 20 shows the thermodynamic equilibrium temperaturecomputed for different values of the equivalence ratio. The rangeof temperatures accessed in the experiments is added in the figurefor the /G ¼ 0:58 and /G ¼ 0:77 cases. The maximal temperaturereached in the experiment is 150 K and 350 K below the equilib-rium states for run 4 and run 2, respectively. This comparison im-plies that in these simulations, the impact of the prescription of theboundary conditions is of importance, since heat losses are signif-icant. Wall temperature measurements were not reported, but theextrapolation of the measured radial temperature profiles from thechamber gives an estimate of the wall temperature in the range of1200� 200 K. Based on this estimate, we prescribed a constantwall temperature of 1200 K for all simulations. By invoking thelow temperature of the combustion process and the relativelylow global residence times found in the configuration, radiation ef-fects were neglected in the following simulations.

The DFPV table was discretized with 151� 15� 121� 11� 11grid points in the direction eZ � SZ � C � �a� �b. The generation ofthe DFPV table, including the computation of the flamelets, tookapproximately 5 h on 24 processors.

5.3. Flow field topology

The high momentum of the air stream inlet promotes the gen-eration of a reverse flow, which advects hot burnt gases toward the

Fig. 20. Thermochemical equilibrium temperature as a function of the mixtureequivalence ratio / under Verissimo et al. [16] operating conditions. Arrowsrepresent the range of measured temperature in the experimental data, andsymbols indicate the mean temperature at the exit of the combustion chamber.




fresh reactants. The two operating conditions impact directly thediluent composition, which contains large amounts of oxygen forrun 4. A comparison of instantaneous temperature fields, obtainedunder the two operating conditions, is shown in Fig. 21. In bothcases, low-velocity regions, localized for x 2 ½0;50�mm and

Fig. 22. Comparison of statistical results at different axial locations in the burner for themole fractions of O2, CO2 and CO. Symbols: experimental data points; continuous line: D

(b)(a)Fig. 21. Instantaneous (left) and mean (right) temperature fields in a median planefor the two operating conditions using the DFPV model. Spatial dimensions are inunits of meters. (a) Run 4; (b) run 2.


r 2 ½20;50�mm, exhibit burned gas temperatures lower than atthe chamber exit. This is attributed to the heat transfer to the wallsand to the long residence times found in these regions. For run 4, ahigh-mean-temperature region (>800 K) can be located at theinterface between the oxidizer stream and burned gases mixedwith methane. For run 2, the mean temperature distribution ismore homogeneous and the temperature is higher than in run 4due to its global equivalence ratio, which is closer to stoichiometry.The instantaneous distributions of run 2 present localized andintermittent high-temperature spots (>1900 K). The mean behav-ior of the two cases is similar to that observed experimentally.

5.4. Statistical flow-field results for run 4: /G ¼ 0:58

Radial profiles of temperature and dry species mole fractionsare given in Fig. 22 for run 4. Experimental results are shown bysymbols. The predicted temperature, obtained from the two DFPVapproaches, is in good agreement with measurements. The FPVmodel overestimates the temperature by 400 K throughout thecombustion chamber. This shows the importance of heat-lossand dilution effects. Temperature predictions that are obtainedfrom both DFPV models exhibit moderate discrepancies (<100 K)for radial position r > 5 mm. By computing unsteady flameletssubject to radiation, comparison between the characteristic radia-tion time [31] and the global residence time of the configurationshowed that temperature overpredictions can be attributed tothe neglect of radiation effects.

Predictions of major species mole fractions are in good agree-ment for all models. Contrary to the temperature distributions, ma-jor species are less sensitive to heat losses, which explains thesimilar results. However, discrepancies can be observed along thecenterline of the configuration, for x 2 ½45;113� mm and r 2 ½0;5�mm. The mole fraction of O2 is underestimated, while CO2 isoverestimated.

Similarly to the results presented in Section 4, the predictions ofcarbon monoxide feature very different evolutions depending onthe combustion model. In the experimental data, the CO spreadswidely in radial direction at the first few probe stations (from 45

operating conditions of run 4, showing (from top to bottom) temperature, and dryFPV model; dashed-dot-dot line: sDFPV model; dashed line: FPV model.



Fig. 23. Comparison of statistical results at different axial locations in the burner for the operating conditions of run 2, showing (from top to bottom) temperature, and drymole fractions of O2, CO2, and CO. Symbols: experimental data points; continuous line: DFPV model; dashed-dot-dot line: sDFPV model; dashed line: FPV model.


to 147 mm). The DFPV model is the only model able to reproducethe correct order of magnitude for CO, even if the predictions arenot satisfactory from 79 to 147 mm. Yet Fig. 22 provides evidenceof the importance of the product gas dissociation and heat losses.More specifically for pollutant emissions, the observed differencesbetween the DFPV models can be attributed to the prescription ofthe b parameter in the sDFPV model, which implies that severalchemical trajectories are not considered (see A.2 for furtherdetails).

5.5. Statistical flow-field results for run 2: /G ¼ 0:77

Radial profiles of temperature and dry species mole fractions forrun 2, corresponding to a MILD combustion regime, are comparedwith experimental data in Fig. 23. Similarly to run 4, the FPV modelwithout consideration of dilution and heat losses overpredictstemperature and CO profiles. Temperature predictions from theDFPV and the sDFPV models are very similar, and in good agree-ment with experimental data. It is encouraging to note that bothdeveloped models converge toward the same equilibrium (probestations located at x 2 ½181 mm; 310 mm�). However, a slightunderestimation of temperature is noticeable along the axis ofsymmetry for regions located between 45 and 79 mm. Away fromthe core region (r > 5 mm), the DFPV models predict higher tem-peratures than seen in the experiments. Computations of unsteadylaminar flamelets subject to radiation for these operating condi-tions suggest that this might be due to the neglect of radiation.However, these discrepancies might also be sensitive to the pre-scribed wall temperature, for which no experimental data areavailable. Yet these models demonstrate their capacity to repro-duce heat loss effects.

Predictions for major species are in better agreement. Discrep-ancies arise in the region x 2 ½11;113�mm and r 2 ½5;15�mm, forwhich O2 is under-predicted and CO2 is overestimated. A similardiscrepancy between temperature and major species was also ob-served for run 4. These discrepancies may be attributed to themodeling hypothesis on the diluent composition, or to experimen-tal uncertainties. Predictions of CO are consistent with


temperature and major species estimate. The missprediction ofthe CO mole fraction at the first measurement station can beexplained by the sensitivity of CO to the temperature field.

Differences between results from the DFPV and the sDFPV mod-els are very low for these operating conditions. For run 2, the DFPVmodel relaxes toward the sDFPV approach, indicating that for thisspecific condition the combustion process is most likely controlledby heat losses rather than dilution effects. Nevertheless, both DFPVapproaches are able to adequately reproduce the first moments ofspecies and temperature field for both conventional and MILDcombustion regimes.

6. Conclusions

Modeling challenges of MILD combustion arise from the pro-nounced dilution of reactants, which induces compositional varia-tions. The dilution influences the reaction chemistry and modelsshould therefore be able to account for these effects. Moreover,most practical combustor configurations operate under nonadia-batic conditions, which requires the consideration of wall heat-losseffects to adequately describe the flame stabilization, fuel conver-sion, and emissions.

In this contribution, a turbulent combustion model was pro-posed that extends a tabulated chemistry model to account forproduct gas dilution and heat-loss effects. Simplifying assumptionswere formulated to decrease the model complexity. Two tabula-tion approaches that extend the FPV model were introduced: theDFPV model, where dilution and heat losses may evolve on differ-ent spatiotemporal scales, and the sDFPV model, where both phe-nomena are constrained. To investigate effects of dilution and heatlosses on the flame structure, a laminar flame-response analysiswas performed. Results from this analysis emphasize the signifi-cance of dilution in the nonequilibrium chemistry and flamestabilization.

A turbulent closure model was proposed, using a presumed PDFmethod to account for turbulence chemistry coupling. An exchangemodel for transferring reaction products into the dilution streamwas formulated. This model was then applied in large eddy




simulations, and two different combustor configurations were con-sidered. In the first case, results obtained with the DFPV modelwere in good agreement with experimental data. It was shown thatthe consideration of dilution effects leads to considerable improve-ments in the CO prediction, demonstrating that the thermochemi-cal trajectory is influenced by dilution. In the second application, acombustor configuration was considered that is operated in theconventional and the MILD operating regime. Overall, the simula-tion results obtained with the proposed model are in reasonableagreement, demonstrating the ability of the model to capture ef-fects of species dilution and heat losses on the flow-field structureand species composition. Future improvements in the model re-quire the consideration of radiation effects and the generalizationof the dilution exchange, which is currently represented by anempirical closure model.

The proposed tabulated chemistry method assumes that the in-ner structure of the turbulent flame front is close to a laminarflame. This assumption remains to be further investigated forlow-temperature flames. Indeed, when the chemical activities slowdown, turbulent eddies penetrate the flame front and may affectthe chemical trajectories. Modeling this phenomenon is a challeng-ing problem.

Acknowledgments

The authors acknowledge funding by ADEME. MI acknowledgesfinancial support through the Office of Naval Research under GrantN00014-10-1-0561. This work was performed using HPC resourcesfrom GENCI-{CCRT/CINES/IDRIS} (Grant 2012-2b0164). Theauthors gratefully acknowledge Dr. V. Moureau for providing theYALES2 code and Dr. M. Costa for providing the experimental data.

Fig. A.1. Schematic representation of the different constraints imposed on temperaturflames; left: DFPV model; right: sDFPV model.

(a)Fig. A.2. Flamelet solutions for Z ¼ ZG (/ ¼ 0:58) and a diluted reactant streams tempsolutions have been illustrated. (a) Reaction rate of progress variable; (b) CO mass fract


Appendix A. Simplified DFPV approach with correlated dilutionand heat-loss evolution

A.1. Model formulation

A simplified DFPV tabulation approach, similar to the one devel-oped by Abtahizadeh et al. [28], can be obtained by assuming thatthe temperature of the diluent is fixed (b is therefore assumed con-stant in the entire combustor). A schematic of this approach isillustrated in Fig. A.1. This assumption constrains heat losses tothe dilution rate and implies that both phenomena evolve similarlyin space and time. When the diluent temperature is fixed, the reac-tion chemistry is then tabulated as a function of three parameters:

w ¼ GbwðZ;Yc;aÞ: ðA:1Þ

In the following, this model is referred to as the ‘‘sDFPV’’ model.The validity of this simplifying assumption is assessed in anapplication to a practical configurations in Section 5. In this model,heat losses and dilution effects are linked, so that HDil is constantand a reduces to

a ¼ H � ZðHFa¼0 � HOx

a¼0Þ � HOxa¼0

HDil � ZGðHFa¼0 � HOx

a¼0Þ � HOxa¼0

: ðA:2Þ

In the context of LES, the filtered relation for a is directlyexpressed from Eq. A.2,

a ¼eH � eZðHF

a¼0 � HOxa¼0Þ � HOx

a¼0

HDil � ZGðHFa¼0 � HOx

a¼0Þ � HOxa¼0

; ðA:3Þ

and this model requires only information about the enthalpy eH andthe mixture fraction eZ .

e and species of the boundary conditions used to generate counterflow diffusion

(b)erature of 1775 K for different values of a and b. For the sake of clarity, only threeion.




A.2. A priori comparison between DFPV and sDFPV models

It is noteworthy that the sDFPV approach depends on thechosen value of b. A preliminary analysis is done to assess the sen-sitivity of the flame responses when different values of the heatloss parameter are applied. To analyze the importance of neglectedtrajectories in the sDFPV model, the study of the chemical trajecto-ries giving the same diluted reactant streams temperature is ofinterest. By prescribing a temperature, at a given equivalence ratio,an iterative sorting procedure can be employed to recover the cor-responding (a; b) parameter pair.

In the following, the mixture fraction is fixed to Z ¼ ZG for therun 4 investigated in Section 5.1, and the diluted reactant streamstemperature is fixed to 1775 K, which corresponds to the maximalmean temperature obtained in this case. Figure A.2(a) illustratesthree solutions of the sorting procedure and presents the reactionrate _xYc as a function of Yc . The DFPV model accounts for all solu-tions, whereas the sDPFV model allows only for a single trajectory.The strongest reaction zone is shifted toward high values of Yc forincreasing values of the diluent temperature. As these trajectoriesare strongly dependent on the parameterization (a; b), the reactionzone can consequently be poorly predicted by the sDFPV model ifnonequilibrium chemistry effects are important.

The influence of the diluent temperature on species evolutionsis highlighted in Fig. A.2(b). Even if the equilibrium mass fractionsare very similar, very different trajectories are found. Therefore, thearbitrary prescription of the value of b required in the sDFPV modelcan have a dramatic influence on the combustion process and pol-lutant formation.

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